We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_p$-weighted $L^p$-space ${\left(L_w^p\left(\Omega \right)\right)}^n$ for any domain $\Omega$ in ${ℝ}^n$, $n\in \mathbb{Z}$, $n\ge 2$, $1<p<\infty$ and Muckenhoupt $A_p$-weight $w\in A_p$. Set $p^{\text{'}}:=p/(p-1)$ and $w^{\text{'}}:=w^{-1/(p-1)}$. Then the Helmholtz decomposition of ${\left(L_w^p\left(\Omega \right)\right)}^n$ and ${\left(L_{w^{\text{'}}}^{p^{\text{'}}}\left(\Omega \right)\right)}^n$ and the variational estimate of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ are equivalent. Furthermore, we can replace $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ by $L_{w,\sigma }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\sigma }^{p^{\text{'}}}\left(\Omega \right)$, respectively. The proof is based on the reflexivity and orthogonality of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w,\sigma }^p\left(\Omega \right)$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with...

In this paper, we present and study a mixed variational method in order to approximate, with the finite element method, a Stokes problem with Tresca friction boundary conditions. These non-linear boundary conditions arise in the modeling of mold filling process by polymer melt, which can slip on a solid wall. The mixed formulation is based on a dualization of the non-differentiable term which define the slip conditions. Existence and uniqueness of both continuous and discrete solutions of these...

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional case....

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional...

The objective of this work is to obtain theoretical estimates on the large and small scales for geophysical flows. Firstly, we consider the shallow water problem in the one-dimensional case, then in the two-dimensional case. Finally we consider geophysical flows under the hydrostatic hypothesis and the Boussinesq approximation. Scale separation is based on Fourier series, with N models in each spatial direction, and the choice of a cut-off level N1 &lt; N to define large and small scales. We...

Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^q$-estimates of second order derivatives uniformly in the angular and translational velocities, ω and...

We formulate sufficient conditions for regularity up to the boundary of a weak solution v in a subdomain Ω × (t₁,t₂) of the time-space cylinder Ω × (0,T) by means of requirements on one of the eigenvalues of the rate of deformation tensor. We assume that Ω is a cube.

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci.16 (2006) 415–438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model...

In this paper we present a theory describing the diffusion limited evaporation of sessile water droplets in presence of contact angle hysteresis. Theory describes two stages of evaporation process: (I) evaporation with a constant radius of the droplet base; and (II) evaporation with constant contact angle. During stage (I) the contact angle decreases from static advancing contact angle to static receding contact angle, during stage (II) the contact...

A small vicinity of a contact line, with well-defined (micro)scales (henceforth the “microstructure”), is studied theoretically for a system of a perfectly wetting liquid, its pure vapor and a superheated flat substrate. At one end, the microstructure terminates in a non-evaporating microfilm owing to the disjoining-pressure-induced Kelvin effect. At the other end, for motionless contact lines, it terminates in a constant film slope (apparent contact...